Optimal. Leaf size=33 \[ \frac {x}{e^x-25 x^2 \left (-4+\frac {x (-x+\log (x))}{4 \log (\log (16))}\right )^2} \]
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Rubi [F]
time = 19.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (-800 x^5+2000 x^6+\left (800 x^4-3200 x^5\right ) \log (x)+1200 x^4 \log ^2(x)\right ) \log ^2(\log (16))+\left (-12800 x^3+38400 x^4-25600 x^3 \log (x)\right ) \log ^3(\log (16))+\left (e^x (256-256 x)+102400 x^2\right ) \log ^4(\log (16))}{625 x^{12}-2500 x^{11} \log (x)+3750 x^{10} \log ^2(x)-2500 x^9 \log ^3(x)+625 x^8 \log ^4(x)+\left (40000 x^{10}-120000 x^9 \log (x)+120000 x^8 \log ^2(x)-40000 x^7 \log ^3(x)\right ) \log (\log (16))+\left (-800 e^x x^6+960000 x^8+\left (1600 e^x x^5-1920000 x^7\right ) \log (x)+\left (-800 e^x x^4+960000 x^6\right ) \log ^2(x)\right ) \log ^2(\log (16))+\left (-25600 e^x x^4+10240000 x^6+\left (25600 e^x x^3-10240000 x^5\right ) \log (x)\right ) \log ^3(\log (16))+\left (256 e^{2 x}-204800 e^x x^2+40960000 x^4\right ) \log ^4(\log (16))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 \log ^2(\log (16)) \left (-50 x^5+125 x^6+75 x^4 \log ^2(x)-800 x^3 \log (\log (16))+2400 x^4 \log (\log (16))+16 e^x \log ^2(\log (16))-16 e^x x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))-50 x^3 \log (x) \left (-x+4 x^2+32 \log (\log (16))\right )\right )}{\left (25 x^6+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))-50 \log (x) \left (x^5+16 x^3 \log (\log (16))\right )\right )^2} \, dx\\ &=\left (16 \log ^2(\log (16))\right ) \int \frac {-50 x^5+125 x^6+75 x^4 \log ^2(x)-800 x^3 \log (\log (16))+2400 x^4 \log (\log (16))+16 e^x \log ^2(\log (16))-16 e^x x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))-50 x^3 \log (x) \left (-x+4 x^2+32 \log (\log (16))\right )}{\left (25 x^6+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))-50 \log (x) \left (x^5+16 x^3 \log (\log (16))\right )\right )^2} \, dx\\ &=\left (16 \log ^2(\log (16))\right ) \int \left (\frac {-1+x}{25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))}+\frac {25 x^2 \left (6 x^4-x^5-10 x^3 \log (x)+2 x^4 \log (x)+4 x^2 \log ^2(x)-x^3 \log ^2(x)+128 x^2 \log (\log (16))-96 x \log (x) \log (\log (16))+512 \log ^2(\log (16))-32 x \log (\log (16)) (1+8 \log (\log (16)))-2 x^3 (1+16 \log (\log (16)))+2 x^2 \log (x) (1+16 \log (\log (16)))\right )}{\left (25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))\right )^2}\right ) \, dx\\ &=\left (16 \log ^2(\log (16))\right ) \int \frac {-1+x}{25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))} \, dx+\left (400 \log ^2(\log (16))\right ) \int \frac {x^2 \left (6 x^4-x^5-10 x^3 \log (x)+2 x^4 \log (x)+4 x^2 \log ^2(x)-x^3 \log ^2(x)+128 x^2 \log (\log (16))-96 x \log (x) \log (\log (16))+512 \log ^2(\log (16))-32 x \log (\log (16)) (1+8 \log (\log (16)))-2 x^3 (1+16 \log (\log (16)))+2 x^2 \log (x) (1+16 \log (\log (16)))\right )}{\left (25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
time = 0.11, size = 70, normalized size = 2.12 \begin {gather*} -\frac {16 x \log ^2(\log (16))}{25 x^6-50 x^5 \log (x)+25 x^4 \log ^2(x)+800 x^4 \log (\log (16))-800 x^3 \log (x) \log (\log (16))-16 e^x \log ^2(\log (16))+6400 x^2 \log ^2(\log (16))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs.
\(2(32)=64\).
time = 0.31, size = 136, normalized size = 4.12
method | result | size |
risch | \(-\frac {16 x \left (4 \ln \left (2\right )^{2}+4 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right )}{25 x^{6}-50 x^{5} \ln \left (x \right )+25 x^{4} \ln \left (x \right )^{2}+1600 x^{4} \ln \left (2\right )-1600 x^{3} \ln \left (x \right ) \ln \left (2\right )+800 \ln \left (\ln \left (2\right )\right ) x^{4}-800 \ln \left (x \right ) \ln \left (\ln \left (2\right )\right ) x^{3}+25600 x^{2} \ln \left (2\right )^{2}+25600 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right ) x^{2}+6400 x^{2} \ln \left (\ln \left (2\right )\right )^{2}-64 \ln \left (2\right )^{2} {\mathrm e}^{x}-64 \,{\mathrm e}^{x} \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-16 \,{\mathrm e}^{x} \ln \left (\ln \left (2\right )\right )^{2}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (34) = 68\).
time = 0.67, size = 120, normalized size = 3.64 \begin {gather*} -\frac {16 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x}{25 \, x^{6} + 25 \, x^{4} \log \left (x\right )^{2} + 800 \, x^{4} {\left (2 \, \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 6400 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} x^{2} - 16 \, {\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (2\right )\right ) + \log \left (\log \left (2\right )\right )^{2}\right )} e^{x} - 50 \, {\left (x^{5} + 16 \, x^{3} {\left (2 \, \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )}\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (34) = 68\).
time = 0.38, size = 71, normalized size = 2.15 \begin {gather*} -\frac {16 \, x \log \left (4 \, \log \left (2\right )\right )^{2}}{25 \, x^{6} - 50 \, x^{5} \log \left (x\right ) + 25 \, x^{4} \log \left (x\right )^{2} + 16 \, {\left (400 \, x^{2} - e^{x}\right )} \log \left (4 \, \log \left (2\right )\right )^{2} + 800 \, {\left (x^{4} - x^{3} \log \left (x\right )\right )} \log \left (4 \, \log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (29) = 58\).
time = 1.24, size = 162, normalized size = 4.91 \begin {gather*} \frac {64 x \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 16 x \log {\left (\log {\left (2 \right )} \right )}^{2} + 64 x \log {\left (2 \right )}^{2}}{- 25 x^{6} + 50 x^{5} \log {\left (x \right )} - 25 x^{4} \log {\left (x \right )}^{2} - 1600 x^{4} \log {\left (2 \right )} - 800 x^{4} \log {\left (\log {\left (2 \right )} \right )} + 800 x^{3} \log {\left (x \right )} \log {\left (\log {\left (2 \right )} \right )} + 1600 x^{3} \log {\left (2 \right )} \log {\left (x \right )} - 25600 x^{2} \log {\left (2 \right )}^{2} - 6400 x^{2} \log {\left (\log {\left (2 \right )} \right )}^{2} - 25600 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \left (64 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 16 \log {\left (\log {\left (2 \right )} \right )}^{2} + 64 \log {\left (2 \right )}^{2}\right ) e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\ln \left (4\,\ln \left (2\right )\right )}^4\,\left ({\mathrm {e}}^x\,\left (256\,x-256\right )-102400\,x^2\right )-{\ln \left (4\,\ln \left (2\right )\right )}^2\,\left (\ln \left (x\right )\,\left (800\,x^4-3200\,x^5\right )+1200\,x^4\,{\ln \left (x\right )}^2-800\,x^5+2000\,x^6\right )+{\ln \left (4\,\ln \left (2\right )\right )}^3\,\left (25600\,x^3\,\ln \left (x\right )+12800\,x^3-38400\,x^4\right )}{{\ln \left (4\,\ln \left (2\right )\right )}^4\,\left (256\,{\mathrm {e}}^{2\,x}-204800\,x^2\,{\mathrm {e}}^x+40960000\,x^4\right )-\ln \left (4\,\ln \left (2\right )\right )\,\left (-40000\,x^{10}+120000\,x^9\,\ln \left (x\right )-120000\,x^8\,{\ln \left (x\right )}^2+40000\,x^7\,{\ln \left (x\right )}^3\right )-2500\,x^{11}\,\ln \left (x\right )+625\,x^8\,{\ln \left (x\right )}^4-2500\,x^9\,{\ln \left (x\right )}^3+3750\,x^{10}\,{\ln \left (x\right )}^2+{\ln \left (4\,\ln \left (2\right )\right )}^3\,\left (10240000\,x^6-25600\,x^4\,{\mathrm {e}}^x+\ln \left (x\right )\,\left (25600\,x^3\,{\mathrm {e}}^x-10240000\,x^5\right )\right )+625\,x^{12}-{\ln \left (4\,\ln \left (2\right )\right )}^2\,\left (800\,x^6\,{\mathrm {e}}^x+{\ln \left (x\right )}^2\,\left (800\,x^4\,{\mathrm {e}}^x-960000\,x^6\right )-960000\,x^8-\ln \left (x\right )\,\left (1600\,x^5\,{\mathrm {e}}^x-1920000\,x^7\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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